Convergence Analysis of Perturbation Theory with Divergent Quantities

[mhill]

Hi, i am stuck at this problem , let be the divergent quantity

$$m= clog(\epsilon) +a_{0}+a_{1}g\epsilon ^{-1}+a_{2}g\epsilon ^{-2} +a_{3}g\epsilon ^{-3}+...+$$

where epsilon tends to 0 and g is just some coupling constant and c ,a_n are real numbers.

then i use the Borel transform of the function $$F(t)= \sum_{n=0}^{\infty}a_{n} \frac{t^{n}}{n!}$$ in this case

$$m= clog(\epsilon)+ \int_{0}^{\infty}dtF(t/\epsilon)e^{-t}$$

my question is, does this last expression have only 2 divergent quantities ? , mainly the one due to log(e) and the second involving the poles of $$F(t/\epsilon)$$

 

[eljose79]

Hello,

Thank you for reaching out for help with this problem. It seems like you have made some progress by using the Borel transform to rewrite the expression in terms of an integral. However, in order to determine the number of divergent quantities in the expression, we need to take a closer look at the behavior of the integral.

First, let's consider the term involving the logarithm. As epsilon approaches 0, the logarithm term will become more and more negative, which means it will diverge to negative infinity. This is a type of divergence known as a logarithmic divergence.

Next, we need to look at the behavior of the integral as t approaches infinity. This will depend on the behavior of F(t/epsilon) as t approaches infinity. If F(t/epsilon) has poles (points where the function becomes infinite), then the integral will also have divergent behavior as t approaches infinity. This is known as a pole divergence.

So, in summary, the expression has two divergent quantities: a logarithmic divergence as epsilon approaches 0, and a pole divergence as t approaches infinity. It is important to note that the number of divergent quantities may change depending on the specific values of c, a_n, and g in the expression.

I hope this helps to clarify the situation. Keep up the good work in trying to solve this problem!

 

[denian]

The convergence analysis of perturbation theory with divergent quantities is a complex and challenging problem in theoretical physics. It is important to carefully analyze the behavior of these divergent quantities in order to understand the limitations and uncertainties in our theoretical calculations.

In this particular case, the expression for m involves a divergent quantity, log(\epsilon), as epsilon tends to 0. This can be problematic as it can lead to non-physical results and unreliable predictions. However, by using the Borel transform of the function F(t), we can rewrite the expression for m in terms of an integral, which may help to better understand the behavior of the divergent quantity.

The integral form of m suggests that there may be only two dominant divergent quantities, log(\epsilon) and the poles of F(t/\epsilon). However, it is important to carefully analyze the behavior of the integral at small values of \epsilon to fully understand the convergence properties of the perturbation theory.

Furthermore, it is important to note that the Borel transform method is just one approach to dealing with divergent quantities in perturbation theory. Other methods, such as renormalization, may also be necessary to fully address the issue of divergences.

In conclusion, the convergence analysis of perturbation theory with divergent quantities is a challenging and ongoing research topic in theoretical physics. Further analysis and investigation are needed to fully understand the behavior of these divergent quantities and their impact on our theoretical predictions.